109:. For example, CTL can specify that when some initial condition is satisfied (e.g., all program variables are positive or no cars on a highway straddle two lanes), then all possible executions of a program avoid some undesirable condition (e.g., dividing a number by zero or two cars colliding on a highway). In this example, the safety property could be verified by a model checker that explores all possible transitions out of program states satisfying the initial condition and ensures that all such executions satisfy the property. Computation tree logic belongs to a class of
22:
2676:
455:
4366:
4056:
2361:
179:
4062:
3752:
2671:{\displaystyle {\bigg (}({\mathcal {M}},s)\models \phi _{1}\Leftrightarrow \phi _{2}{\bigg )}\Leftrightarrow {\bigg (}{\Big (}{\big (}({\mathcal {M}},s)\models \phi _{1}{\big )}\land {\big (}({\mathcal {M}},s)\models \phi _{2}{\big )}{\Big )}\lor {\Big (}\neg {\big (}({\mathcal {M}},s)\models \phi _{1}{\big )}\land \neg {\big (}({\mathcal {M}},s)\models \phi _{2}{\big )}{\Big )}{\bigg )}}
450:{\displaystyle {\begin{aligned}\phi &::=\bot \mid \top \mid p\mid (\neg \phi )\mid (\phi \land \phi )\mid (\phi \lor \phi )\mid (\phi \Rightarrow \phi )\mid (\phi \Leftrightarrow \phi )\\&\mid \quad {\mbox{AX }}\phi \mid {\mbox{EX }}\phi \mid {\mbox{AF }}\phi \mid {\mbox{EF }}\phi \mid {\mbox{AG }}\phi \mid {\mbox{EG }}\phi \mid {\mbox{A }}\mid {\mbox{E }}\end{aligned}}}
2355:
3746:
3554:
3362:
3170:
1977:
2166:
2978:
2827:
4361:{\displaystyle {\bigg (}({\mathcal {M}},s)\models E{\bigg )}\Leftrightarrow {\bigg (}\exists \langle s_{1}\rightarrow s_{2}\rightarrow \ldots \rangle (s=s_{1})\exists i{\Big (}{\big (}({\mathcal {M}},s_{i})\models \phi _{2}{\big )}\land {\big (}\forall (j<i)({\mathcal {M}},s_{j})\models \phi _{1}{\big )}{\Big )}{\bigg )}}
4051:{\displaystyle {\bigg (}({\mathcal {M}},s)\models A{\bigg )}\Leftrightarrow {\bigg (}\forall \langle s_{1}\rightarrow s_{2}\rightarrow \ldots \rangle (s=s_{1})\exists i{\Big (}{\big (}({\mathcal {M}},s_{i})\models \phi _{2}{\big )}\land {\big (}\forall (j<i)({\mathcal {M}},s_{j})\models \phi _{1}{\big )}{\Big )}{\bigg )}}
2172:
3560:
3368:
3176:
2984:
5307:"Depending on what happens in the future (E), it's possible that for the rest of time (G), I'll be guaranteed at least one (AF) chocolate-liking day still ahead of me. However, if something ever goes wrong, then all bets are off and there's no guarantee about whether I'll ever like chocolate."
1794:
152:
Since the introduction of CTL, there has been debate about the relative merits of CTL and LTL. Because CTL is more computationally efficient to model check, it has become more common in industrial use, and many of the most successful model-checking tools use CTL as a specification language.
1983:
1788:
1594:
2833:
2682:
5335:"From now until it's warm outside, I will like chocolate every single day. Once it's warm outside, all bets are off as to whether I'll like chocolate anymore. Oh, and it's guaranteed to be warm outside eventually, even if only for a single day."
1686:
2350:{\displaystyle {\Big (}({\mathcal {M}},s)\models \phi _{1}\Rightarrow \phi _{2}{\Big )}\Leftrightarrow {\Big (}{\big (}({\mathcal {M}},s)\not \models \phi _{1}{\big )}\lor {\big (}({\mathcal {M}},s)\models \phi _{2}{\big )}{\Big )}}
3741:{\displaystyle {\Big (}({\mathcal {M}},s)\models EF\phi {\Big )}\Leftrightarrow {\Big (}\exists \langle s_{1}\rightarrow s_{2}\rightarrow \ldots \rangle (s=s_{1})\exists i{\big (}({\mathcal {M}},s_{i})\models \phi {\big )}{\Big )}}
3549:{\displaystyle {\Big (}({\mathcal {M}},s)\models AF\phi {\Big )}\Leftrightarrow {\Big (}\forall \langle s_{1}\rightarrow s_{2}\rightarrow \ldots \rangle (s=s_{1})\exists i{\big (}({\mathcal {M}},s_{i})\models \phi {\big )}{\Big )}}
3357:{\displaystyle {\Big (}({\mathcal {M}},s)\models EG\phi {\Big )}\Leftrightarrow {\Big (}\exists \langle s_{1}\rightarrow s_{2}\rightarrow \ldots \rangle (s=s_{1})\forall i{\big (}({\mathcal {M}},s_{i})\models \phi {\big )}{\Big )}}
3165:{\displaystyle {\Big (}({\mathcal {M}},s)\models AG\phi {\Big )}\Leftrightarrow {\Big (}\forall \langle s_{1}\rightarrow s_{2}\rightarrow \ldots \rangle (s=s_{1})\forall i{\big (}({\mathcal {M}},s_{i})\models \phi {\big )}{\Big )}}
5708:
4376:
Rules 10–15 above refer to computation paths in models and are what ultimately characterise the "Computation Tree"; they are assertions about the nature of the infinitely deep computation tree rooted at the given state
1972:{\displaystyle {\Big (}({\mathcal {M}},s)\models \phi _{1}\land \phi _{2}{\Big )}\Leftrightarrow {\Big (}{\big (}({\mathcal {M}},s)\models \phi _{1}{\big )}\land {\big (}({\mathcal {M}},s)\models \phi _{2}{\big )}{\Big )}}
2161:{\displaystyle {\Big (}({\mathcal {M}},s)\models \phi _{1}\lor \phi _{2}{\Big )}\Leftrightarrow {\Big (}{\big (}({\mathcal {M}},s)\models \phi _{1}{\big )}\lor {\big (}({\mathcal {M}},s)\models \phi _{2}{\big )}{\Big )}}
663:
544:
4557:
1692:
720:
1199:
184:
5234:
5153:
4970:
4919:
5072:
5021:
1321:
1253:
5507:
A reduction from the model-checking problem of QCTL with the structure semantics, to TQBF (true quantified
Boolean formulae) has been proposed, in order to take advantage of the QBF solvers.
1501:
4748:
4703:
4658:
2973:{\displaystyle {\Big (}({\mathcal {M}},s)\models EX\phi {\Big )}\Leftrightarrow {\Big (}\exists \langle s\rightarrow s_{1}\rangle {\big (}({\mathcal {M}},s_{1})\models \phi {\big )}{\Big )}}
2822:{\displaystyle {\Big (}({\mathcal {M}},s)\models AX\phi {\Big )}\Leftrightarrow {\Big (}\forall \langle s\rightarrow s_{1}\rangle {\big (}({\mathcal {M}},s_{1})\models \phi {\big )}{\Big )}}
1473:
4469:
117: (LTL). Although there are properties expressible only in CTL and properties expressible only in LTL, all properties expressible in either logic can also be expressed in
1425:
4859:
4604:
4582:
4515:
4493:
789:
767:
745:
1600:
1373:
1005:, the temporal operators can be freely mixed. In CTL, operators must always be grouped in pairs: one path operator followed by a state operator. See the examples below.
600:
571:
5474:
5451:
4812:
1347:
4443:
4423:
1493:
97:
structure in which the future is not determined; there are different paths in the future, any one of which might be an actual path that is realized. It is used in
5286:"It's always possible (AF) that I will suddenly start liking chocolate for the rest of time." (Note: not just the rest of my life, since my life is finite, while
4774:
4395:
1393:
1273:
1219:
478:
5803:
5311:
The two following examples show the difference between CTL and CTL*, as they allow for the until operator to not be qualified with any path operator (
5360:"It's possible that: there will eventually come a time when it will be warm forever (AG.Q) and that before that time there will always be
615:
487:
4520:
1783:{\displaystyle {\Big (}({\mathcal {M}},s)\models \neg \phi {\Big )}\Leftrightarrow {\Big (}({\mathcal {M}},s)\not \models \phi {\Big )}}
823:
are the usual ones: ¬, ∨, ∧, ⇒ and ⇔. Along with these operators CTL formulas can also make use of the boolean constants
798:
as its building blocks to make statements about the states of a system. These propositions are then combined into formulas using
5766:
674:
1017:
In CTL there are minimal sets of operators. All CTL formulas can be transformed to use only those operators. This is useful in
4445:
are said to be semantically equivalent if any state in any model that satisfies one also satisfies the other. This is denoted
5847:
5750:
5684:
5641:
5566:
1589:{\displaystyle {\Big (}({\mathcal {M}},s)\models \top {\Big )}\land {\Big (}({\mathcal {M}},s)\not \models \bot {\Big )}}
1154:
5159:
5078:
4925:
4874:
5878:
5027:
4976:
1278:
1224:
5731:
4709:
4664:
4619:
1436:
65:
43:
36:
5888:
5378:
4753:
It can be shown using such identities that a subset of the CTL temporal connectives is adequate if it contains
106:
5521:
5614:
David, Amélie; Laroussinie, Francois; Markey, Nicolas (2016). Desharnais, Josée; Jagadeesan, Radha (eds.).
5492:
5657:
Hossain, Akash; Laroussinie, François (2019). Gamper, Johann; Pinchinat, Sophie; Sciavicco, Guido (eds.).
5600:
5485:
4448:
5780:
5588:
5547:
1681:{\displaystyle {\Big (}({\mathcal {M}},s)\models p{\Big )}\Leftrightarrow {\Big (}p\in L(s){\Big )}}
1406:
30:
4817:
4587:
4565:
4498:
4476:
772:
750:
728:
5883:
1255:
is a transition relation, assumed to be serial, i.e. every state has at least one successor, and
5392:
are not equivalent and they have a common subset, which is a proper subset of both CTL and LTL.
1352:
795:
581:
552:
5799:"Automatic verification of finite-state concurrent systems using temporal logic specifications"
5775:
5456:
5433:
4779:
47:
5526:
5389:
5385:
114:
4868:; they allow unfolding the verification of a CTL connective towards its successors in time.
1326:
4428:
4408:
1478:
170:
5764:(1985). "Decision procedures and expressiveness in the temporal logic of branching time".
8:
5500:
4610:
1396:
863:
xists: there exists at least one path starting from the current state where Φ holds.
166:
98:
94:
4756:
5822:
5709:"Design and synthesis of synchronisation skeletons using branching time temporal logic"
5690:
5576:
5427:
4380:
1378:
1258:
1204:
820:
463:
146:
86:
5553:. Lecture Notes in Computer Science. Vol. 2031. Springer, Berlin. pp. 1–22.
4517:
are duals, being universal and existential computation path quantifiers respectively:
5843:
5789:
5746:
5727:
5694:
5680:
5658:
5637:
5562:
5516:
5374:
1148:
5826:
5615:
5812:
5785:
5719:
5670:
5669:. Dagstuhl, Germany: Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik: 11:1–11:20.
5627:
5626:. Dagstuhl, Germany: Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik: 28:1–28:15.
5554:
799:
546:
comprises a complete set of connectives, and the others can be defined using them.
162:
134:
130:
5632:
5835:
5663:
26th
International Symposium on Temporal Representation and Reasoning (TIME 2019)
5496:
1400:
5863:
5675:
5761:
1018:
803:
481:
110:
101:
of software or hardware artifacts, typically by software applications known as
5872:
5716:
Logic of
Programs, Proceedings of Workshop, Lecture Notes in Computer Science
828:
102:
5558:
5484:
the tree semantics. We label nodes of the computation tree. QCTL* = QCTL =
1275:
is a labelling function, assigning propositional letters to states. Let
5796:
5723:
1431:
484:. It is not necessary to use all connectives – for example,
5817:
5798:
5269:"It's possible I may like chocolate some day, at least for one day."
853:
ll: Φ has to hold on all paths starting from the current state.
658:{\displaystyle {\mbox{EF }}({\mbox{EG }}p\Rightarrow {\mbox{AF }}r)}
539:{\displaystyle \{\neg ,\land ,{\mbox{AX}},{\mbox{AU}},{\mbox{EU}}\}}
5373:
Computation tree logic (CTL) is a subset of CTL* as well as of the
4552:{\displaystyle \neg \mathrm {A} \Phi \equiv \mathrm {E} \neg \Phi }
5491:
the structure semantics. We label states. QCTL* = QCTL = MSO over
5620:
27th
International Conference on Concurrency Theory (CONCUR 2016)
5488:
over trees. Model checking and satisfiability are tower complete.
5244:
Let "P" mean "I like chocolate" and Q mean "It's warm outside."
885:
has to hold at the next state (this operator is sometimes noted
5665:. Leibniz International Proceedings in Informatics (LIPIcs).
5622:. Leibniz International Proceedings in Informatics (LIPIcs).
1036:
Some of the transformations used for temporal operators are:
824:
5759:
5745:(Second ed.). Cambridge University Press. p. 207.
5377:. CTL is also a fragment of Alur, Henzinger and Kupferman's
5256:"I will like chocolate from now on, no matter what happens."
1006:
1002:
118:
90:
5797:
Clarke, E. M.; Emerson, E. A. & Sistla, A. P. (1986).
927:
eventually has to hold (somewhere on the subsequent path).
715:{\displaystyle {\mbox{EF }}{\big (}r{\mbox{ U }}q{\big )}}
609:
For example, the following is a well-formed CTL formula:
5706:
5834:
Emerson, E. A. (1990). "Temporal and modal logic". In
5613:
1021:. One minimal set of operators is: {true, ∨, ¬,
696:
679:
643:
630:
620:
586:
557:
527:
517:
507:
431:
418:
402:
389:
376:
363:
350:
337:
324:
311:
5804:
5740:
5459:
5436:
5364:
way to get me to like chocolate the next day (EX.P)."
5162:
5081:
5030:
4979:
4928:
4877:
4820:
4782:
4759:
4712:
4667:
4622:
4590:
4568:
4523:
4501:
4479:
4451:
4431:
4411:
4383:
4065:
3755:
3563:
3371:
3179:
2987:
2836:
2685:
2364:
2175:
1986:
1797:
1695:
1603:
1504:
1481:
1439:
1409:
1381:
1355:
1329:
1281:
1261:
1227:
1207:
1157:
775:
753:
731:
677:
618:
584:
555:
490:
466:
182:
5718:. Vol. 131. Springer, Berlin. pp. 52–71.
1194:{\displaystyle {\mathcal {M}}=(S,{\rightarrow },L)}
5656:
5468:
5445:
5229:{\displaystyle E\equiv \psi \lor (\phi \land EXE)}
5228:
5148:{\displaystyle A\equiv \psi \lor (\phi \land AXA)}
5147:
5066:
5015:
4965:{\displaystyle EG\phi \equiv \phi \land EXEG\phi }
4964:
4914:{\displaystyle AG\phi \equiv \phi \land AXAG\phi }
4913:
4853:
4806:
4768:
4742:
4697:
4652:
4598:
4576:
4551:
4509:
4487:
4463:
4437:
4417:
4389:
4360:
4050:
3740:
3548:
3356:
3164:
2972:
2821:
2670:
2349:
2160:
1971:
1782:
1680:
1588:
1487:
1467:
1419:
1387:
1367:
1341:
1315:
1267:
1247:
1213:
1193:
783:
761:
739:
714:
657:
594:
565:
538:
472:
449:
5067:{\displaystyle EF\phi \equiv \phi \lor EXEF\phi }
5016:{\displaystyle AF\phi \equiv \phi \lor AXAF\phi }
4353:
4346:
4212:
4139:
4129:
4068:
4043:
4036:
3902:
3829:
3819:
3758:
3733:
3614:
3604:
3566:
3541:
3422:
3412:
3374:
3349:
3230:
3220:
3182:
3157:
3038:
3028:
2990:
2965:
2887:
2877:
2839:
2814:
2736:
2726:
2688:
2663:
2656:
2548:
2538:
2436:
2429:
2419:
2367:
2342:
2240:
2230:
2178:
2153:
2051:
2041:
1989:
1964:
1862:
1852:
1800:
1775:
1743:
1733:
1698:
1673:
1648:
1638:
1606:
1581:
1549:
1539:
1507:
1316:{\displaystyle {\mathcal {M}}=(S,\rightarrow ,L)}
1248:{\displaystyle {\rightarrow }\subseteq S\times S}
5870:
5840:Handbook of Theoretical Computer Science, vol. B
4864:The important equivalences below are called the
668:The following is not a well-formed CTL formula:
105:, which determine if a given artifact possesses
4743:{\displaystyle \neg AX\phi \equiv EX\neg \phi }
4698:{\displaystyle \neg EF\phi \equiv AG\neg \phi }
4653:{\displaystyle \neg AF\phi \equiv EG\neg \phi }
1468:{\displaystyle ({\mathcal {M}},s\models \phi )}
602:means 'along at least (there Exists) one path'
5833:
5368:
4339:
4275:
4265:
4219:
4029:
3965:
3955:
3909:
3726:
3687:
3534:
3495:
3342:
3303:
3150:
3111:
2958:
2919:
2807:
2768:
2649:
2610:
2597:
2558:
2531:
2492:
2482:
2443:
2335:
2296:
2286:
2247:
2146:
2107:
2097:
2058:
1957:
1918:
1908:
1869:
707:
687:
137:in 1981, who used it to synthesize so-called
4848:
4821:
4801:
4783:
4179:
4147:
3869:
3837:
3654:
3622:
3462:
3430:
3270:
3238:
3078:
3046:
2914:
2895:
2763:
2744:
533:
491:
5548:"Branching vs. Linear Time: Final Showdown"
1012:
4371:
910:has to hold on the entire subsequent path.
839:The temporal operators are the following:
5816:
5779:
5674:
5631:
5388:(LTL) are both a subset of CTL*. CTL and
66:Learn how and when to remove this message
4400:
29:This article includes a list of general
5767:Journal of Computer and System Sciences
5871:
1009:is strictly more expressive than CTL.
995:operator is sometimes called "unless".
169:for CTL is generated by the following
5545:
834:
725:The problem with this string is that
814:
15:
5478:quantified computational tree logic
1151:. A transition system is a triple
1137:
987:is that there is no guarantee that
747:can occur only when paired with an
13:
5707:E.M. Clarke; E.A. Emerson (1981).
5460:
5437:
4734:
4713:
4689:
4668:
4644:
4623:
4592:
4570:
4546:
4543:
4539:
4532:
4528:
4524:
4503:
4481:
4303:
4280:
4229:
4204:
4144:
4078:
3993:
3970:
3919:
3894:
3834:
3768:
3697:
3679:
3619:
3576:
3505:
3487:
3427:
3384:
3313:
3295:
3235:
3192:
3121:
3103:
3043:
3000:
2929:
2892:
2849:
2778:
2741:
2698:
2620:
2605:
2568:
2553:
2502:
2453:
2377:
2306:
2257:
2188:
2117:
2068:
1999:
1928:
1879:
1810:
1753:
1725:
1708:
1616:
1576:
1559:
1534:
1517:
1445:
1412:
1284:
1160:
1147:CTL formulae are interpreted over
777:
755:
733:
494:
218:
203:
197:
35:it lacks sufficient corresponding
14:
5900:
5857:
5480:(QCTL). There are two semantics:
5384:Computation tree logic (CTL) and
4464:{\displaystyle \phi \equiv \psi }
1323:be such a transition model, with
5842:. MIT Press. pp. 955–1072.
5741:Michael Huth; Mark Ryan (2004).
5399:.P exists in LTL but not in CTL.
156:
20:
5616:"On the Expressiveness of QCTL"
5417:.P exist in CTL but not in LTL.
5379:alternating-time temporal logic
959:will be verified in the future.
309:
5650:
5607:
5539:
5223:
5220:
5208:
5190:
5178:
5166:
5142:
5139:
5127:
5109:
5097:
5085:
4321:
4298:
4295:
4283:
4247:
4224:
4201:
4182:
4173:
4160:
4134:
4124:
4098:
4089:
4073:
4011:
3988:
3985:
3973:
3937:
3914:
3891:
3872:
3863:
3850:
3824:
3814:
3788:
3779:
3763:
3715:
3692:
3676:
3657:
3648:
3635:
3609:
3587:
3571:
3523:
3500:
3484:
3465:
3456:
3443:
3417:
3395:
3379:
3331:
3308:
3292:
3273:
3264:
3251:
3225:
3203:
3187:
3139:
3116:
3100:
3081:
3072:
3059:
3033:
3011:
2995:
2947:
2924:
2901:
2882:
2860:
2844:
2796:
2773:
2750:
2731:
2709:
2693:
2631:
2615:
2579:
2563:
2513:
2497:
2464:
2448:
2424:
2404:
2388:
2372:
2317:
2301:
2268:
2252:
2235:
2215:
2199:
2183:
2128:
2112:
2079:
2063:
2046:
2010:
1994:
1939:
1923:
1890:
1874:
1857:
1821:
1805:
1764:
1748:
1738:
1719:
1703:
1668:
1662:
1643:
1627:
1611:
1570:
1554:
1528:
1512:
1462:
1440:
1430:Then the relation of semantic
1420:{\displaystyle {\mathcal {M}}}
1310:
1301:
1292:
1229:
1188:
1178:
1168:
652:
639:
626:
440:
424:
411:
395:
296:
290:
284:
278:
272:
266:
260:
248:
242:
230:
224:
215:
1:
5633:10.4230/LIPIcs.CONCUR.2016.28
5532:
5522:Fair computational tree logic
5421:
4861:and the boolean connectives.
1142:
107:safety or liveness properties
5790:10.1016/0022-0000(85)90001-7
5659:"From Quantified CTL to QBF"
4854:{\displaystyle \{EG,AF,AU\}}
4599:{\displaystyle \mathrm {F} }
4577:{\displaystyle \mathrm {G} }
4510:{\displaystyle \mathrm {E} }
4488:{\displaystyle \mathrm {A} }
809:
784:{\displaystyle \mathrm {E} }
762:{\displaystyle \mathrm {A} }
740:{\displaystyle \mathrm {U} }
89:, meaning that its model of
7:
5676:10.4230/LIPIcs.TIME.2019.11
5510:
5426:CTL has been extended with
5369:Relations with other logics
5239:
991:will ever be verified. The
983:holds. The difference with
10:
5905:
4613:can be formulated in CTL:
1475:is defined recursively on
1368:{\displaystyle \phi \in F}
868:Path-specific quantifiers
595:{\displaystyle {\mbox{E}}}
566:{\displaystyle {\mbox{A}}}
129:CTL was first proposed by
124:
5879:Logic in computer science
5743:Logic in Computer Science
5469:{\displaystyle \forall p}
5446:{\displaystyle \exists p}
4807:{\displaystyle \{AX,EX\}}
955:holds. This implies that
139:synchronisation skeletons
5546:Vardi, Moshe Y. (2001).
1013:Minimal set of operators
573:means 'along All paths'
5559:10.1007/3-540-45319-9_1
4372:Characterisation of CTL
951:until at some position
843:Quantifiers over paths
50:more precise citations.
5889:Automata (computation)
5864:Teaching slides of CTL
5499:but satisfiability is
5470:
5447:
5230:
5149:
5068:
5017:
4966:
4915:
4855:
4808:
4770:
4744:
4699:
4654:
4600:
4578:
4553:
4511:
4489:
4465:
4439:
4419:
4391:
4362:
4052:
3742:
3550:
3358:
3166:
2974:
2823:
2672:
2351:
2162:
1973:
1784:
1682:
1590:
1489:
1469:
1421:
1389:
1369:
1343:
1342:{\displaystyle s\in S}
1317:
1269:
1249:
1215:
1195:
785:
763:
741:
716:
659:
596:
567:
540:
474:
451:
85:) is a branching-time
79:Computation tree logic
5527:Linear temporal logic
5471:
5448:
5386:linear temporal logic
5231:
5150:
5069:
5018:
4967:
4916:
4856:
4809:
4771:
4745:
4700:
4655:
4609:Hence an instance of
4601:
4579:
4554:
4512:
4490:
4466:
4440:
4438:{\displaystyle \psi }
4420:
4418:{\displaystyle \phi }
4401:Semantic equivalences
4392:
4363:
4053:
3743:
3551:
3359:
3167:
2975:
2824:
2673:
2352:
2163:
1974:
1785:
1683:
1591:
1490:
1488:{\displaystyle \phi }
1470:
1422:
1390:
1370:
1344:
1318:
1270:
1250:
1216:
1196:
786:
764:
742:
717:
660:
597:
568:
541:
480:ranges over a set of
475:
452:
115:linear temporal logic
5495:. Model checking is
5457:
5434:
5160:
5079:
5028:
4977:
4926:
4875:
4818:
4814:and at least one of
4780:
4757:
4710:
4665:
4620:
4588:
4566:
4562:Furthermore, so are
4521:
4499:
4477:
4473:It can be seen that
4449:
4429:
4409:
4381:
4063:
3753:
3561:
3369:
3177:
2985:
2834:
2683:
2362:
2173:
1984:
1795:
1693:
1601:
1502:
1479:
1437:
1407:
1397:well-formed formulas
1379:
1353:
1327:
1279:
1259:
1225:
1221:is a set of states,
1205:
1155:
859:Φ –
849:Φ –
773:
751:
729:
675:
616:
582:
553:
488:
464:
180:
167:well-formed formulas
796:atomic propositions
147:concurrent programs
99:formal verification
5724:10.1007/BFb0025774
5466:
5443:
5226:
5145:
5064:
5013:
4962:
4911:
4851:
4804:
4776:, at least one of
4769:{\displaystyle EU}
4766:
4740:
4695:
4650:
4596:
4574:
4549:
4507:
4485:
4461:
4435:
4415:
4387:
4358:
4048:
3738:
3546:
3354:
3162:
2970:
2819:
2668:
2347:
2158:
1969:
1780:
1678:
1586:
1485:
1465:
1417:
1385:
1365:
1339:
1313:
1265:
1245:
1211:
1191:
1149:transition systems
979:has to hold until
835:Temporal operators
804:temporal operators
781:
759:
737:
712:
700:
683:
655:
647:
634:
624:
592:
590:
563:
561:
536:
531:
521:
511:
470:
447:
445:
435:
422:
406:
393:
380:
367:
354:
341:
328:
315:
5849:978-0-262-22039-2
5818:10.1145/5397.5399
5752:978-0-521-54310-1
5686:978-3-95977-127-6
5643:978-3-95977-017-0
5595:Missing or empty
5568:978-3-540-41865-8
5517:Probabilistic CTL
4390:{\displaystyle s}
1388:{\displaystyle F}
1268:{\displaystyle L}
1214:{\displaystyle S}
821:logical operators
815:Logical operators
800:logical operators
699:
682:
646:
633:
623:
589:
560:
530:
520:
510:
473:{\displaystyle p}
434:
421:
405:
392:
379:
366:
353:
340:
327:
314:
76:
75:
68:
5896:
5853:
5830:
5820:
5793:
5783:
5760:Emerson, E. A.;
5756:
5737:
5713:
5699:
5698:
5678:
5654:
5648:
5647:
5635:
5611:
5605:
5604:
5598:
5592:
5586:
5582:
5580:
5572:
5552:
5543:
5475:
5473:
5472:
5467:
5452:
5450:
5449:
5444:
5375:modal μ calculus
5235:
5233:
5232:
5227:
5154:
5152:
5151:
5146:
5073:
5071:
5070:
5065:
5022:
5020:
5019:
5014:
4971:
4969:
4968:
4963:
4920:
4918:
4917:
4912:
4860:
4858:
4857:
4852:
4813:
4811:
4810:
4805:
4775:
4773:
4772:
4767:
4749:
4747:
4746:
4741:
4704:
4702:
4701:
4696:
4659:
4657:
4656:
4651:
4611:De Morgan's laws
4605:
4603:
4602:
4597:
4595:
4583:
4581:
4580:
4575:
4573:
4558:
4556:
4555:
4550:
4542:
4531:
4516:
4514:
4513:
4508:
4506:
4494:
4492:
4491:
4486:
4484:
4470:
4468:
4467:
4462:
4444:
4442:
4441:
4436:
4424:
4422:
4421:
4416:
4396:
4394:
4393:
4388:
4367:
4365:
4364:
4359:
4357:
4356:
4350:
4349:
4343:
4342:
4336:
4335:
4320:
4319:
4307:
4306:
4279:
4278:
4269:
4268:
4262:
4261:
4246:
4245:
4233:
4232:
4223:
4222:
4216:
4215:
4200:
4199:
4172:
4171:
4159:
4158:
4143:
4142:
4133:
4132:
4123:
4122:
4110:
4109:
4082:
4081:
4072:
4071:
4057:
4055:
4054:
4049:
4047:
4046:
4040:
4039:
4033:
4032:
4026:
4025:
4010:
4009:
3997:
3996:
3969:
3968:
3959:
3958:
3952:
3951:
3936:
3935:
3923:
3922:
3913:
3912:
3906:
3905:
3890:
3889:
3862:
3861:
3849:
3848:
3833:
3832:
3823:
3822:
3813:
3812:
3800:
3799:
3772:
3771:
3762:
3761:
3747:
3745:
3744:
3739:
3737:
3736:
3730:
3729:
3714:
3713:
3701:
3700:
3691:
3690:
3675:
3674:
3647:
3646:
3634:
3633:
3618:
3617:
3608:
3607:
3580:
3579:
3570:
3569:
3555:
3553:
3552:
3547:
3545:
3544:
3538:
3537:
3522:
3521:
3509:
3508:
3499:
3498:
3483:
3482:
3455:
3454:
3442:
3441:
3426:
3425:
3416:
3415:
3388:
3387:
3378:
3377:
3363:
3361:
3360:
3355:
3353:
3352:
3346:
3345:
3330:
3329:
3317:
3316:
3307:
3306:
3291:
3290:
3263:
3262:
3250:
3249:
3234:
3233:
3224:
3223:
3196:
3195:
3186:
3185:
3171:
3169:
3168:
3163:
3161:
3160:
3154:
3153:
3138:
3137:
3125:
3124:
3115:
3114:
3099:
3098:
3071:
3070:
3058:
3057:
3042:
3041:
3032:
3031:
3004:
3003:
2994:
2993:
2979:
2977:
2976:
2971:
2969:
2968:
2962:
2961:
2946:
2945:
2933:
2932:
2923:
2922:
2913:
2912:
2891:
2890:
2881:
2880:
2853:
2852:
2843:
2842:
2828:
2826:
2825:
2820:
2818:
2817:
2811:
2810:
2795:
2794:
2782:
2781:
2772:
2771:
2762:
2761:
2740:
2739:
2730:
2729:
2702:
2701:
2692:
2691:
2677:
2675:
2674:
2669:
2667:
2666:
2660:
2659:
2653:
2652:
2646:
2645:
2624:
2623:
2614:
2613:
2601:
2600:
2594:
2593:
2572:
2571:
2562:
2561:
2552:
2551:
2542:
2541:
2535:
2534:
2528:
2527:
2506:
2505:
2496:
2495:
2486:
2485:
2479:
2478:
2457:
2456:
2447:
2446:
2440:
2439:
2433:
2432:
2423:
2422:
2416:
2415:
2403:
2402:
2381:
2380:
2371:
2370:
2356:
2354:
2353:
2348:
2346:
2345:
2339:
2338:
2332:
2331:
2310:
2309:
2300:
2299:
2290:
2289:
2283:
2282:
2261:
2260:
2251:
2250:
2244:
2243:
2234:
2233:
2227:
2226:
2214:
2213:
2192:
2191:
2182:
2181:
2167:
2165:
2164:
2159:
2157:
2156:
2150:
2149:
2143:
2142:
2121:
2120:
2111:
2110:
2101:
2100:
2094:
2093:
2072:
2071:
2062:
2061:
2055:
2054:
2045:
2044:
2038:
2037:
2025:
2024:
2003:
2002:
1993:
1992:
1978:
1976:
1975:
1970:
1968:
1967:
1961:
1960:
1954:
1953:
1932:
1931:
1922:
1921:
1912:
1911:
1905:
1904:
1883:
1882:
1873:
1872:
1866:
1865:
1856:
1855:
1849:
1848:
1836:
1835:
1814:
1813:
1804:
1803:
1789:
1787:
1786:
1781:
1779:
1778:
1757:
1756:
1747:
1746:
1737:
1736:
1712:
1711:
1702:
1701:
1687:
1685:
1684:
1679:
1677:
1676:
1652:
1651:
1642:
1641:
1620:
1619:
1610:
1609:
1595:
1593:
1592:
1587:
1585:
1584:
1563:
1562:
1553:
1552:
1543:
1542:
1521:
1520:
1511:
1510:
1494:
1492:
1491:
1486:
1474:
1472:
1471:
1466:
1449:
1448:
1426:
1424:
1423:
1418:
1416:
1415:
1394:
1392:
1391:
1386:
1374:
1372:
1371:
1366:
1348:
1346:
1345:
1340:
1322:
1320:
1319:
1314:
1288:
1287:
1274:
1272:
1271:
1266:
1254:
1252:
1251:
1246:
1232:
1220:
1218:
1217:
1212:
1200:
1198:
1197:
1192:
1181:
1164:
1163:
1138:Semantics of CTL
877: – Ne
790:
788:
787:
782:
780:
768:
766:
765:
760:
758:
746:
744:
743:
738:
736:
721:
719:
718:
713:
711:
710:
701:
697:
691:
690:
684:
680:
664:
662:
661:
656:
648:
644:
635:
631:
625:
621:
601:
599:
598:
593:
591:
587:
572:
570:
569:
564:
562:
558:
545:
543:
542:
537:
532:
528:
522:
518:
512:
508:
479:
477:
476:
471:
456:
454:
453:
448:
446:
436:
432:
423:
419:
407:
403:
394:
390:
381:
377:
368:
364:
355:
351:
342:
338:
329:
325:
316:
312:
302:
145:abstractions of
135:E. Allen Emerson
131:Edmund M. Clarke
71:
64:
60:
57:
51:
46:this article by
37:inline citations
24:
23:
16:
5904:
5903:
5899:
5898:
5897:
5895:
5894:
5893:
5869:
5868:
5860:
5850:
5836:Jan van Leeuwen
5781:10.1.1.221.6187
5753:
5734:
5711:
5703:
5702:
5687:
5655:
5651:
5644:
5612:
5608:
5596:
5594:
5584:
5583:
5574:
5573:
5569:
5550:
5544:
5540:
5535:
5513:
5497:PSPACE-complete
5458:
5455:
5454:
5435:
5432:
5431:
5430:quantification
5424:
5371:
5242:
5161:
5158:
5157:
5080:
5077:
5076:
5029:
5026:
5025:
4978:
4975:
4974:
4927:
4924:
4923:
4876:
4873:
4872:
4819:
4816:
4815:
4781:
4778:
4777:
4758:
4755:
4754:
4711:
4708:
4707:
4666:
4663:
4662:
4621:
4618:
4617:
4591:
4589:
4586:
4585:
4569:
4567:
4564:
4563:
4538:
4527:
4522:
4519:
4518:
4502:
4500:
4497:
4496:
4480:
4478:
4475:
4474:
4450:
4447:
4446:
4430:
4427:
4426:
4410:
4407:
4406:
4403:
4382:
4379:
4378:
4374:
4352:
4351:
4345:
4344:
4338:
4337:
4331:
4327:
4315:
4311:
4302:
4301:
4274:
4273:
4264:
4263:
4257:
4253:
4241:
4237:
4228:
4227:
4218:
4217:
4211:
4210:
4195:
4191:
4167:
4163:
4154:
4150:
4138:
4137:
4128:
4127:
4118:
4114:
4105:
4101:
4077:
4076:
4067:
4066:
4064:
4061:
4060:
4042:
4041:
4035:
4034:
4028:
4027:
4021:
4017:
4005:
4001:
3992:
3991:
3964:
3963:
3954:
3953:
3947:
3943:
3931:
3927:
3918:
3917:
3908:
3907:
3901:
3900:
3885:
3881:
3857:
3853:
3844:
3840:
3828:
3827:
3818:
3817:
3808:
3804:
3795:
3791:
3767:
3766:
3757:
3756:
3754:
3751:
3750:
3732:
3731:
3725:
3724:
3709:
3705:
3696:
3695:
3686:
3685:
3670:
3666:
3642:
3638:
3629:
3625:
3613:
3612:
3603:
3602:
3575:
3574:
3565:
3564:
3562:
3559:
3558:
3540:
3539:
3533:
3532:
3517:
3513:
3504:
3503:
3494:
3493:
3478:
3474:
3450:
3446:
3437:
3433:
3421:
3420:
3411:
3410:
3383:
3382:
3373:
3372:
3370:
3367:
3366:
3348:
3347:
3341:
3340:
3325:
3321:
3312:
3311:
3302:
3301:
3286:
3282:
3258:
3254:
3245:
3241:
3229:
3228:
3219:
3218:
3191:
3190:
3181:
3180:
3178:
3175:
3174:
3156:
3155:
3149:
3148:
3133:
3129:
3120:
3119:
3110:
3109:
3094:
3090:
3066:
3062:
3053:
3049:
3037:
3036:
3027:
3026:
2999:
2998:
2989:
2988:
2986:
2983:
2982:
2964:
2963:
2957:
2956:
2941:
2937:
2928:
2927:
2918:
2917:
2908:
2904:
2886:
2885:
2876:
2875:
2848:
2847:
2838:
2837:
2835:
2832:
2831:
2813:
2812:
2806:
2805:
2790:
2786:
2777:
2776:
2767:
2766:
2757:
2753:
2735:
2734:
2725:
2724:
2697:
2696:
2687:
2686:
2684:
2681:
2680:
2662:
2661:
2655:
2654:
2648:
2647:
2641:
2637:
2619:
2618:
2609:
2608:
2596:
2595:
2589:
2585:
2567:
2566:
2557:
2556:
2547:
2546:
2537:
2536:
2530:
2529:
2523:
2519:
2501:
2500:
2491:
2490:
2481:
2480:
2474:
2470:
2452:
2451:
2442:
2441:
2435:
2434:
2428:
2427:
2418:
2417:
2411:
2407:
2398:
2394:
2376:
2375:
2366:
2365:
2363:
2360:
2359:
2341:
2340:
2334:
2333:
2327:
2323:
2305:
2304:
2295:
2294:
2285:
2284:
2278:
2274:
2256:
2255:
2246:
2245:
2239:
2238:
2229:
2228:
2222:
2218:
2209:
2205:
2187:
2186:
2177:
2176:
2174:
2171:
2170:
2152:
2151:
2145:
2144:
2138:
2134:
2116:
2115:
2106:
2105:
2096:
2095:
2089:
2085:
2067:
2066:
2057:
2056:
2050:
2049:
2040:
2039:
2033:
2029:
2020:
2016:
1998:
1997:
1988:
1987:
1985:
1982:
1981:
1963:
1962:
1956:
1955:
1949:
1945:
1927:
1926:
1917:
1916:
1907:
1906:
1900:
1896:
1878:
1877:
1868:
1867:
1861:
1860:
1851:
1850:
1844:
1840:
1831:
1827:
1809:
1808:
1799:
1798:
1796:
1793:
1792:
1774:
1773:
1752:
1751:
1742:
1741:
1732:
1731:
1707:
1706:
1697:
1696:
1694:
1691:
1690:
1672:
1671:
1647:
1646:
1637:
1636:
1615:
1614:
1605:
1604:
1602:
1599:
1598:
1580:
1579:
1558:
1557:
1548:
1547:
1538:
1537:
1516:
1515:
1506:
1505:
1503:
1500:
1499:
1480:
1477:
1476:
1444:
1443:
1438:
1435:
1434:
1411:
1410:
1408:
1405:
1404:
1380:
1377:
1376:
1354:
1351:
1350:
1328:
1325:
1324:
1283:
1282:
1280:
1277:
1276:
1260:
1257:
1256:
1228:
1226:
1223:
1222:
1206:
1203:
1202:
1177:
1159:
1158:
1156:
1153:
1152:
1145:
1140:
1015:
837:
817:
812:
776:
774:
771:
770:
754:
752:
749:
748:
732:
730:
727:
726:
706:
705:
695:
686:
685:
678:
676:
673:
672:
642:
629:
619:
617:
614:
613:
585:
583:
580:
579:
556:
554:
551:
550:
526:
516:
506:
489:
486:
485:
482:atomic formulas
465:
462:
461:
444:
443:
430:
417:
401:
388:
375:
362:
349:
336:
323:
310:
300:
299:
190:
183:
181:
178:
177:
159:
127:
111:temporal logics
72:
61:
55:
52:
42:Please help to
41:
25:
21:
12:
11:
5:
5902:
5892:
5891:
5886:
5884:Temporal logic
5881:
5867:
5866:
5859:
5858:External links
5856:
5855:
5854:
5848:
5831:
5811:(2): 244–263.
5794:
5762:Halpern, J. Y.
5757:
5751:
5738:
5732:
5701:
5700:
5685:
5649:
5642:
5606:
5585:|journal=
5567:
5537:
5536:
5534:
5531:
5530:
5529:
5524:
5519:
5512:
5509:
5505:
5504:
5489:
5465:
5462:
5442:
5439:
5423:
5420:
5419:
5418:
5400:
5370:
5367:
5366:
5365:
5357:
5356:
5337:
5336:
5332:
5331:
5309:
5308:
5304:
5303:
5292:
5291:
5283:
5282:
5271:
5270:
5266:
5265:
5258:
5257:
5253:
5252:
5241:
5238:
5237:
5236:
5225:
5222:
5219:
5216:
5213:
5210:
5207:
5204:
5201:
5198:
5195:
5192:
5189:
5186:
5183:
5180:
5177:
5174:
5171:
5168:
5165:
5155:
5144:
5141:
5138:
5135:
5132:
5129:
5126:
5123:
5120:
5117:
5114:
5111:
5108:
5105:
5102:
5099:
5096:
5093:
5090:
5087:
5084:
5074:
5063:
5060:
5057:
5054:
5051:
5048:
5045:
5042:
5039:
5036:
5033:
5023:
5012:
5009:
5006:
5003:
5000:
4997:
4994:
4991:
4988:
4985:
4982:
4972:
4961:
4958:
4955:
4952:
4949:
4946:
4943:
4940:
4937:
4934:
4931:
4921:
4910:
4907:
4904:
4901:
4898:
4895:
4892:
4889:
4886:
4883:
4880:
4866:expansion laws
4850:
4847:
4844:
4841:
4838:
4835:
4832:
4829:
4826:
4823:
4803:
4800:
4797:
4794:
4791:
4788:
4785:
4765:
4762:
4751:
4750:
4739:
4736:
4733:
4730:
4727:
4724:
4721:
4718:
4715:
4705:
4694:
4691:
4688:
4685:
4682:
4679:
4676:
4673:
4670:
4660:
4649:
4646:
4643:
4640:
4637:
4634:
4631:
4628:
4625:
4594:
4572:
4548:
4545:
4541:
4537:
4534:
4530:
4526:
4505:
4483:
4460:
4457:
4454:
4434:
4414:
4402:
4399:
4386:
4373:
4370:
4369:
4368:
4355:
4348:
4341:
4334:
4330:
4326:
4323:
4318:
4314:
4310:
4305:
4300:
4297:
4294:
4291:
4288:
4285:
4282:
4277:
4272:
4267:
4260:
4256:
4252:
4249:
4244:
4240:
4236:
4231:
4226:
4221:
4214:
4209:
4206:
4203:
4198:
4194:
4190:
4187:
4184:
4181:
4178:
4175:
4170:
4166:
4162:
4157:
4153:
4149:
4146:
4141:
4136:
4131:
4126:
4121:
4117:
4113:
4108:
4104:
4100:
4097:
4094:
4091:
4088:
4085:
4080:
4075:
4070:
4058:
4045:
4038:
4031:
4024:
4020:
4016:
4013:
4008:
4004:
4000:
3995:
3990:
3987:
3984:
3981:
3978:
3975:
3972:
3967:
3962:
3957:
3950:
3946:
3942:
3939:
3934:
3930:
3926:
3921:
3916:
3911:
3904:
3899:
3896:
3893:
3888:
3884:
3880:
3877:
3874:
3871:
3868:
3865:
3860:
3856:
3852:
3847:
3843:
3839:
3836:
3831:
3826:
3821:
3816:
3811:
3807:
3803:
3798:
3794:
3790:
3787:
3784:
3781:
3778:
3775:
3770:
3765:
3760:
3748:
3735:
3728:
3723:
3720:
3717:
3712:
3708:
3704:
3699:
3694:
3689:
3684:
3681:
3678:
3673:
3669:
3665:
3662:
3659:
3656:
3653:
3650:
3645:
3641:
3637:
3632:
3628:
3624:
3621:
3616:
3611:
3606:
3601:
3598:
3595:
3592:
3589:
3586:
3583:
3578:
3573:
3568:
3556:
3543:
3536:
3531:
3528:
3525:
3520:
3516:
3512:
3507:
3502:
3497:
3492:
3489:
3486:
3481:
3477:
3473:
3470:
3467:
3464:
3461:
3458:
3453:
3449:
3445:
3440:
3436:
3432:
3429:
3424:
3419:
3414:
3409:
3406:
3403:
3400:
3397:
3394:
3391:
3386:
3381:
3376:
3364:
3351:
3344:
3339:
3336:
3333:
3328:
3324:
3320:
3315:
3310:
3305:
3300:
3297:
3294:
3289:
3285:
3281:
3278:
3275:
3272:
3269:
3266:
3261:
3257:
3253:
3248:
3244:
3240:
3237:
3232:
3227:
3222:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3194:
3189:
3184:
3172:
3159:
3152:
3147:
3144:
3141:
3136:
3132:
3128:
3123:
3118:
3113:
3108:
3105:
3102:
3097:
3093:
3089:
3086:
3083:
3080:
3077:
3074:
3069:
3065:
3061:
3056:
3052:
3048:
3045:
3040:
3035:
3030:
3025:
3022:
3019:
3016:
3013:
3010:
3007:
3002:
2997:
2992:
2980:
2967:
2960:
2955:
2952:
2949:
2944:
2940:
2936:
2931:
2926:
2921:
2916:
2911:
2907:
2903:
2900:
2897:
2894:
2889:
2884:
2879:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2851:
2846:
2841:
2829:
2816:
2809:
2804:
2801:
2798:
2793:
2789:
2785:
2780:
2775:
2770:
2765:
2760:
2756:
2752:
2749:
2746:
2743:
2738:
2733:
2728:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2700:
2695:
2690:
2678:
2665:
2658:
2651:
2644:
2640:
2636:
2633:
2630:
2627:
2622:
2617:
2612:
2607:
2604:
2599:
2592:
2588:
2584:
2581:
2578:
2575:
2570:
2565:
2560:
2555:
2550:
2545:
2540:
2533:
2526:
2522:
2518:
2515:
2512:
2509:
2504:
2499:
2494:
2489:
2484:
2477:
2473:
2469:
2466:
2463:
2460:
2455:
2450:
2445:
2438:
2431:
2426:
2421:
2414:
2410:
2406:
2401:
2397:
2393:
2390:
2387:
2384:
2379:
2374:
2369:
2357:
2344:
2337:
2330:
2326:
2322:
2319:
2316:
2313:
2308:
2303:
2298:
2293:
2288:
2281:
2277:
2273:
2270:
2267:
2264:
2259:
2254:
2249:
2242:
2237:
2232:
2225:
2221:
2217:
2212:
2208:
2204:
2201:
2198:
2195:
2190:
2185:
2180:
2168:
2155:
2148:
2141:
2137:
2133:
2130:
2127:
2124:
2119:
2114:
2109:
2104:
2099:
2092:
2088:
2084:
2081:
2078:
2075:
2070:
2065:
2060:
2053:
2048:
2043:
2036:
2032:
2028:
2023:
2019:
2015:
2012:
2009:
2006:
2001:
1996:
1991:
1979:
1966:
1959:
1952:
1948:
1944:
1941:
1938:
1935:
1930:
1925:
1920:
1915:
1910:
1903:
1899:
1895:
1892:
1889:
1886:
1881:
1876:
1871:
1864:
1859:
1854:
1847:
1843:
1839:
1834:
1830:
1826:
1823:
1820:
1817:
1812:
1807:
1802:
1790:
1777:
1772:
1769:
1766:
1763:
1760:
1755:
1750:
1745:
1740:
1735:
1730:
1727:
1724:
1721:
1718:
1715:
1710:
1705:
1700:
1688:
1675:
1670:
1667:
1664:
1661:
1658:
1655:
1650:
1645:
1640:
1635:
1632:
1629:
1626:
1623:
1618:
1613:
1608:
1596:
1583:
1578:
1575:
1572:
1569:
1566:
1561:
1556:
1551:
1546:
1541:
1536:
1533:
1530:
1527:
1524:
1519:
1514:
1509:
1484:
1464:
1461:
1458:
1455:
1452:
1447:
1442:
1414:
1395:is the set of
1384:
1364:
1361:
1358:
1338:
1335:
1332:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1286:
1264:
1244:
1241:
1238:
1235:
1231:
1210:
1190:
1187:
1184:
1180:
1176:
1173:
1170:
1167:
1162:
1144:
1141:
1139:
1136:
1135:
1134:
1116:
1095:
1075:
1058:
1019:model checking
1014:
1011:
999:
998:
997:
996:
971: –
960:
939: –
928:
919: –
911:
902: –
894:
866:
865:
864:
854:
836:
833:
816:
813:
811:
808:
779:
757:
735:
723:
722:
709:
704:
694:
689:
666:
665:
654:
651:
641:
638:
628:
607:
606:
577:
535:
525:
515:
505:
502:
499:
496:
493:
469:
458:
457:
442:
439:
429:
426:
416:
413:
410:
400:
397:
387:
384:
374:
371:
361:
358:
348:
345:
335:
332:
322:
319:
308:
305:
303:
301:
298:
295:
292:
289:
286:
283:
280:
277:
274:
271:
268:
265:
262:
259:
256:
253:
250:
247:
244:
241:
238:
235:
232:
229:
226:
223:
220:
217:
214:
211:
208:
205:
202:
199:
196:
193:
191:
189:
186:
185:
158:
155:
126:
123:
113:that includes
103:model checkers
74:
73:
28:
26:
19:
9:
6:
4:
3:
2:
5901:
5890:
5887:
5885:
5882:
5880:
5877:
5876:
5874:
5865:
5862:
5861:
5851:
5845:
5841:
5837:
5832:
5828:
5824:
5819:
5814:
5810:
5806:
5805:
5800:
5795:
5791:
5787:
5782:
5777:
5773:
5769:
5768:
5763:
5758:
5754:
5748:
5744:
5739:
5735:
5733:3-540-11212-X
5729:
5725:
5721:
5717:
5710:
5705:
5704:
5696:
5692:
5688:
5682:
5677:
5672:
5668:
5664:
5660:
5653:
5645:
5639:
5634:
5629:
5625:
5621:
5617:
5610:
5602:
5590:
5578:
5570:
5564:
5560:
5556:
5549:
5542:
5538:
5528:
5525:
5523:
5520:
5518:
5515:
5514:
5508:
5502:
5498:
5494:
5490:
5487:
5483:
5482:
5481:
5479:
5463:
5440:
5429:
5416:
5412:
5408:
5404:
5401:
5398:
5395:
5394:
5393:
5391:
5387:
5382:
5380:
5376:
5363:
5359:
5358:
5354:
5350:
5346:
5342:
5339:
5338:
5334:
5333:
5329:
5325:
5322:
5321:
5320:
5318:
5314:
5306:
5305:
5301:
5297:
5294:
5293:
5290:is infinite).
5289:
5285:
5284:
5280:
5276:
5273:
5272:
5268:
5267:
5263:
5260:
5259:
5255:
5254:
5250:
5247:
5246:
5245:
5217:
5214:
5211:
5205:
5202:
5199:
5196:
5193:
5187:
5184:
5181:
5175:
5172:
5169:
5163:
5156:
5136:
5133:
5130:
5124:
5121:
5118:
5115:
5112:
5106:
5103:
5100:
5094:
5091:
5088:
5082:
5075:
5061:
5058:
5055:
5052:
5049:
5046:
5043:
5040:
5037:
5034:
5031:
5024:
5010:
5007:
5004:
5001:
4998:
4995:
4992:
4989:
4986:
4983:
4980:
4973:
4959:
4956:
4953:
4950:
4947:
4944:
4941:
4938:
4935:
4932:
4929:
4922:
4908:
4905:
4902:
4899:
4896:
4893:
4890:
4887:
4884:
4881:
4878:
4871:
4870:
4869:
4867:
4862:
4845:
4842:
4839:
4836:
4833:
4830:
4827:
4824:
4798:
4795:
4792:
4789:
4786:
4763:
4760:
4737:
4731:
4728:
4725:
4722:
4719:
4716:
4706:
4692:
4686:
4683:
4680:
4677:
4674:
4671:
4661:
4647:
4641:
4638:
4635:
4632:
4629:
4626:
4616:
4615:
4614:
4612:
4607:
4560:
4535:
4471:
4458:
4455:
4452:
4432:
4412:
4405:The formulae
4398:
4384:
4332:
4328:
4324:
4316:
4312:
4308:
4292:
4289:
4286:
4270:
4258:
4254:
4250:
4242:
4238:
4234:
4207:
4196:
4192:
4188:
4185:
4176:
4168:
4164:
4155:
4151:
4119:
4115:
4111:
4106:
4102:
4095:
4092:
4086:
4083:
4059:
4022:
4018:
4014:
4006:
4002:
3998:
3982:
3979:
3976:
3960:
3948:
3944:
3940:
3932:
3928:
3924:
3897:
3886:
3882:
3878:
3875:
3866:
3858:
3854:
3845:
3841:
3809:
3805:
3801:
3796:
3792:
3785:
3782:
3776:
3773:
3749:
3721:
3718:
3710:
3706:
3702:
3682:
3671:
3667:
3663:
3660:
3651:
3643:
3639:
3630:
3626:
3599:
3596:
3593:
3590:
3584:
3581:
3557:
3529:
3526:
3518:
3514:
3510:
3490:
3479:
3475:
3471:
3468:
3459:
3451:
3447:
3438:
3434:
3407:
3404:
3401:
3398:
3392:
3389:
3365:
3337:
3334:
3326:
3322:
3318:
3298:
3287:
3283:
3279:
3276:
3267:
3259:
3255:
3246:
3242:
3215:
3212:
3209:
3206:
3200:
3197:
3173:
3145:
3142:
3134:
3130:
3126:
3106:
3095:
3091:
3087:
3084:
3075:
3067:
3063:
3054:
3050:
3023:
3020:
3017:
3014:
3008:
3005:
2981:
2953:
2950:
2942:
2938:
2934:
2909:
2905:
2898:
2872:
2869:
2866:
2863:
2857:
2854:
2830:
2802:
2799:
2791:
2787:
2783:
2758:
2754:
2747:
2721:
2718:
2715:
2712:
2706:
2703:
2679:
2642:
2638:
2634:
2628:
2625:
2602:
2590:
2586:
2582:
2576:
2573:
2543:
2524:
2520:
2516:
2510:
2507:
2487:
2475:
2471:
2467:
2461:
2458:
2412:
2408:
2399:
2395:
2391:
2385:
2382:
2358:
2328:
2324:
2320:
2314:
2311:
2291:
2279:
2275:
2271:
2265:
2262:
2223:
2219:
2210:
2206:
2202:
2196:
2193:
2169:
2139:
2135:
2131:
2125:
2122:
2102:
2090:
2086:
2082:
2076:
2073:
2034:
2030:
2026:
2021:
2017:
2013:
2007:
2004:
1980:
1950:
1946:
1942:
1936:
1933:
1913:
1901:
1897:
1893:
1887:
1884:
1845:
1841:
1837:
1832:
1828:
1824:
1818:
1815:
1791:
1770:
1767:
1761:
1758:
1728:
1722:
1716:
1713:
1689:
1665:
1659:
1656:
1653:
1633:
1630:
1624:
1621:
1597:
1573:
1567:
1564:
1544:
1531:
1525:
1522:
1498:
1497:
1496:
1482:
1459:
1456:
1453:
1450:
1433:
1428:
1402:
1398:
1382:
1362:
1359:
1356:
1336:
1333:
1330:
1307:
1304:
1298:
1295:
1289:
1262:
1242:
1239:
1236:
1233:
1208:
1185:
1182:
1174:
1171:
1165:
1150:
1132:
1128:
1124:
1120:
1117:
1114:
1110:
1106:
1102:
1099:
1096:
1094:
1090:
1086:
1082:
1079:
1076:
1073:
1069:
1065:
1062:
1059:
1056:
1053:
1049:
1045:
1042:
1039:
1038:
1037:
1034:
1032:
1028:
1024:
1020:
1010:
1008:
1004:
994:
990:
986:
982:
978:
974:
970:
967:
964:
961:
958:
954:
950:
946:
942:
938:
935:
932:
929:
926:
922:
918:
915:
912:
909:
905:
901:
898:
895:
892:
888:
884:
880:
876:
873:
870:
869:
867:
862:
858:
855:
852:
848:
845:
844:
842:
841:
840:
832:
830:
826:
822:
807:
805:
801:
797:
792:
702:
698: U
692:
671:
670:
669:
649:
636:
612:
611:
610:
605:
578:
576:
549:
548:
547:
523:
513:
503:
500:
497:
483:
467:
437:
433: U
427:
414:
408:
404: U
398:
385:
382:
372:
369:
359:
356:
346:
343:
333:
330:
320:
317:
306:
304:
293:
287:
281:
275:
269:
263:
257:
254:
251:
245:
239:
236:
233:
227:
221:
212:
209:
206:
200:
194:
192:
187:
176:
175:
174:
172:
168:
164:
157:Syntax of CTL
154:
150:
148:
144:
140:
136:
132:
122:
120:
116:
112:
108:
104:
100:
96:
92:
88:
84:
80:
70:
67:
59:
49:
45:
39:
38:
32:
27:
18:
17:
5839:
5808:
5802:
5771:
5765:
5742:
5715:
5666:
5662:
5652:
5623:
5619:
5609:
5597:|title=
5541:
5506:
5477:
5428:second-order
5425:
5414:
5410:
5406:
5402:
5396:
5383:
5372:
5361:
5352:
5348:
5344:
5340:
5327:
5323:
5316:
5312:
5310:
5299:
5295:
5287:
5278:
5274:
5261:
5248:
5243:
4865:
4863:
4752:
4608:
4561:
4472:
4404:
4375:
1429:
1146:
1130:
1126:
1122:
1118:
1112:
1108:
1104:
1100:
1097:
1092:
1088:
1084:
1080:
1077:
1071:
1067:
1063:
1060:
1054:
1051:
1047:
1043:
1040:
1035:
1030:
1026:
1022:
1016:
1000:
992:
988:
984:
980:
976:
972:
968:
965:
962:
956:
952:
948:
947:has to hold
944:
940:
936:
933:
930:
924:
920:
916:
913:
907:
903:
899:
896:
890:
886:
882:
878:
874:
871:
860:
856:
850:
846:
838:
818:
793:
724:
667:
608:
603:
575:(inevitably)
574:
459:
160:
151:
142:
138:
128:
82:
78:
77:
62:
56:October 2015
53:
34:
5774:(1): 1–24.
5501:undecidable
5409:.Q)∧(
975:eak until:
889:instead of
48:introducing
5873:Categories
5533:References
5422:Extensions
5413:¬Q))) and
1432:entailment
1143:Definition
1050:( because
604:(possibly)
31:references
5776:CiteSeerX
5695:195345645
5587:ignored (
5577:cite book
5461:∀
5438:∃
5218:ψ
5212:ϕ
5197:∧
5194:ϕ
5188:∨
5185:ψ
5182:≡
5176:ψ
5170:ϕ
5137:ψ
5131:ϕ
5116:∧
5113:ϕ
5107:∨
5104:ψ
5101:≡
5095:ψ
5089:ϕ
5062:ϕ
5047:∨
5044:ϕ
5041:≡
5038:ϕ
5011:ϕ
4996:∨
4993:ϕ
4990:≡
4987:ϕ
4960:ϕ
4945:∧
4942:ϕ
4939:≡
4936:ϕ
4909:ϕ
4894:∧
4891:ϕ
4888:≡
4885:ϕ
4738:ϕ
4735:¬
4726:≡
4723:ϕ
4714:¬
4693:ϕ
4690:¬
4681:≡
4678:ϕ
4669:¬
4648:ϕ
4645:¬
4636:≡
4633:ϕ
4624:¬
4547:Φ
4544:¬
4536:≡
4533:Φ
4525:¬
4459:ψ
4456:≡
4453:ϕ
4433:ψ
4413:ϕ
4329:ϕ
4325:⊨
4281:∀
4271:∧
4255:ϕ
4251:⊨
4205:∃
4180:⟩
4177:…
4174:→
4161:→
4148:⟨
4145:∃
4135:⇔
4116:ϕ
4103:ϕ
4093:⊨
4019:ϕ
4015:⊨
3971:∀
3961:∧
3945:ϕ
3941:⊨
3895:∃
3870:⟩
3867:…
3864:→
3851:→
3838:⟨
3835:∀
3825:⇔
3806:ϕ
3793:ϕ
3783:⊨
3722:ϕ
3719:⊨
3680:∃
3655:⟩
3652:…
3649:→
3636:→
3623:⟨
3620:∃
3610:⇔
3600:ϕ
3591:⊨
3530:ϕ
3527:⊨
3488:∃
3463:⟩
3460:…
3457:→
3444:→
3431:⟨
3428:∀
3418:⇔
3408:ϕ
3399:⊨
3338:ϕ
3335:⊨
3296:∀
3271:⟩
3268:…
3265:→
3252:→
3239:⟨
3236:∃
3226:⇔
3216:ϕ
3207:⊨
3146:ϕ
3143:⊨
3104:∀
3079:⟩
3076:…
3073:→
3060:→
3047:⟨
3044:∀
3034:⇔
3024:ϕ
3015:⊨
2954:ϕ
2951:⊨
2915:⟩
2902:→
2896:⟨
2893:∃
2883:⇔
2873:ϕ
2864:⊨
2803:ϕ
2800:⊨
2764:⟩
2751:→
2745:⟨
2742:∀
2732:⇔
2722:ϕ
2713:⊨
2639:ϕ
2635:⊨
2606:¬
2603:∧
2587:ϕ
2583:⊨
2554:¬
2544:∨
2521:ϕ
2517:⊨
2488:∧
2472:ϕ
2468:⊨
2425:⇔
2409:ϕ
2405:⇔
2396:ϕ
2392:⊨
2325:ϕ
2321:⊨
2292:∨
2276:ϕ
2236:⇔
2220:ϕ
2216:⇒
2207:ϕ
2203:⊨
2136:ϕ
2132:⊨
2103:∨
2087:ϕ
2083:⊨
2047:⇔
2031:ϕ
2027:∨
2018:ϕ
2014:⊨
1947:ϕ
1943:⊨
1914:∧
1898:ϕ
1894:⊨
1858:⇔
1842:ϕ
1838:∧
1829:ϕ
1825:⊨
1771:ϕ
1739:⇔
1729:ϕ
1726:¬
1723:⊨
1657:∈
1644:⇔
1631:⊨
1577:⊥
1545:∧
1535:⊤
1532:⊨
1483:ϕ
1460:ϕ
1457:⊨
1399:over the
1360:∈
1357:ϕ
1334:∈
1302:→
1240:×
1234:⊆
1230:→
1179:→
906:lobally:
810:Operators
794:CTL uses
640:⇒
501:∧
495:¬
438:ϕ
428:ϕ
415:∣
409:ϕ
399:ϕ
386:∣
383:ϕ
373:∣
370:ϕ
360:∣
357:ϕ
347:∣
344:ϕ
334:∣
331:ϕ
321:∣
318:ϕ
307:∣
294:ϕ
291:⇔
288:ϕ
282:∣
276:ϕ
273:⇒
270:ϕ
264:∣
258:ϕ
255:∨
252:ϕ
246:∣
240:ϕ
237:∧
234:ϕ
228:∣
222:ϕ
219:¬
213:∣
207:∣
204:⊤
201:∣
198:⊥
188:ϕ
95:tree-like
5827:52853200
5511:See also
5240:Examples
2272:⊭
1768:⊭
1574:⊭
1401:language
1375:, where
1201:, where
1125:∨
949:at least
923:inally:
681:EF
645:AF
632:EG
622:EF
378:EG
365:AG
352:EF
339:AF
326:EX
313:AX
163:language
5838:(ed.).
5381:(ATL).
1091:) == ¬
420:E
391:A
171:grammar
125:History
44:improve
5846:
5825:
5778:
5749:
5730:
5693:
5683:
5640:
5565:
5493:graphs
1349:, and
1121:== ¬(
943:ntil:
769:or an
460:where
33:, but
5823:S2CID
5712:(PDF)
5691:S2CID
5551:(PDF)
5415:AG.EF
5405:(P⇒((
1057:== )
829:false
93:is a
87:logic
5844:ISBN
5747:ISBN
5728:ISBN
5681:ISBN
5638:ISBN
5601:help
5589:help
5563:ISBN
5453:and
5362:some
5355:.Q))
4584:and
4495:and
4425:and
4290:<
3980:<
1107:== ¬
1083:== ¬
1066:== ¬
1007:CTL*
1003:CTL*
827:and
825:true
819:The
802:and
161:The
133:and
119:CTL*
91:time
5813:doi
5786:doi
5720:doi
5671:doi
5667:147
5628:doi
5555:doi
5486:MSO
5476:to
5390:LTL
5347:.P)
5319:):
5315:or
1403:of
1133:) )
1103:==
1046:==
1033:}.
1001:In
881:t:
791:.
195:::=
165:of
143:i.e
83:CTL
5875::
5821:.
5807:.
5801:.
5784:.
5772:30
5770:.
5726:.
5714:.
5689:.
5679:.
5661:.
5636:.
5624:59
5618:.
5593:;
5581::
5579:}}
5575:{{
5561:.
5411:EX
5407:EX
5403:AG
5397:FG
5353:AG
5345:EX
5343:((
5341:EF
5330:Q)
5326:(P
5324:AG
5302:.P
5300:AF
5296:EG
5281:.P
5279:EG
5275:AF
5264:.P
5262:EF
5251:.P
5249:AG
4606:.
4559:.
4397:.
1495::
1427:.
1129:(¬
1127:EG
1111:(¬
1109:EG
1098:AF
1087:(¬
1085:EF
1078:AG
1070:(¬
1068:EX
1061:AX
1041:EF
1031:EX
1029:,
1027:EU
1025:,
1023:EG
893:).
831:.
806:.
529:EU
519:AU
509:AX
173::
149:.
141:,
121:.
5852:.
5829:.
5815::
5809:8
5792:.
5788::
5755:.
5736:.
5722::
5697:.
5673::
5646:.
5630::
5603:)
5599:(
5591:)
5571:.
5557::
5503:.
5464:p
5441:p
5351:(
5349:U
5328:U
5317:E
5313:A
5298:.
5288:G
5277:.
5224:)
5221:]
5215:U
5209:[
5206:E
5203:X
5200:E
5191:(
5179:]
5173:U
5167:[
5164:E
5143:)
5140:]
5134:U
5128:[
5125:A
5122:X
5119:A
5110:(
5098:]
5092:U
5086:[
5083:A
5059:F
5056:E
5053:X
5050:E
5035:F
5032:E
5008:F
5005:A
5002:X
4999:A
4984:F
4981:A
4957:G
4954:E
4951:X
4948:E
4933:G
4930:E
4906:G
4903:A
4900:X
4897:A
4882:G
4879:A
4849:}
4846:U
4843:A
4840:,
4837:F
4834:A
4831:,
4828:G
4825:E
4822:{
4802:}
4799:X
4796:E
4793:,
4790:X
4787:A
4784:{
4764:U
4761:E
4732:X
4729:E
4720:X
4717:A
4687:G
4684:A
4675:F
4672:E
4642:G
4639:E
4630:F
4627:A
4593:F
4571:G
4540:E
4529:A
4504:E
4482:A
4385:s
4354:)
4347:)
4340:)
4333:1
4322:)
4317:j
4313:s
4309:,
4304:M
4299:(
4296:)
4293:i
4287:j
4284:(
4276:(
4266:)
4259:2
4248:)
4243:i
4239:s
4235:,
4230:M
4225:(
4220:(
4213:(
4208:i
4202:)
4197:1
4193:s
4189:=
4186:s
4183:(
4169:2
4165:s
4156:1
4152:s
4140:(
4130:)
4125:]
4120:2
4112:U
4107:1
4099:[
4096:E
4090:)
4087:s
4084:,
4079:M
4074:(
4069:(
4044:)
4037:)
4030:)
4023:1
4012:)
4007:j
4003:s
3999:,
3994:M
3989:(
3986:)
3983:i
3977:j
3974:(
3966:(
3956:)
3949:2
3938:)
3933:i
3929:s
3925:,
3920:M
3915:(
3910:(
3903:(
3898:i
3892:)
3887:1
3883:s
3879:=
3876:s
3873:(
3859:2
3855:s
3846:1
3842:s
3830:(
3820:)
3815:]
3810:2
3802:U
3797:1
3789:[
3786:A
3780:)
3777:s
3774:,
3769:M
3764:(
3759:(
3734:)
3727:)
3716:)
3711:i
3707:s
3703:,
3698:M
3693:(
3688:(
3683:i
3677:)
3672:1
3668:s
3664:=
3661:s
3658:(
3644:2
3640:s
3631:1
3627:s
3615:(
3605:)
3597:F
3594:E
3588:)
3585:s
3582:,
3577:M
3572:(
3567:(
3542:)
3535:)
3524:)
3519:i
3515:s
3511:,
3506:M
3501:(
3496:(
3491:i
3485:)
3480:1
3476:s
3472:=
3469:s
3466:(
3452:2
3448:s
3439:1
3435:s
3423:(
3413:)
3405:F
3402:A
3396:)
3393:s
3390:,
3385:M
3380:(
3375:(
3350:)
3343:)
3332:)
3327:i
3323:s
3319:,
3314:M
3309:(
3304:(
3299:i
3293:)
3288:1
3284:s
3280:=
3277:s
3274:(
3260:2
3256:s
3247:1
3243:s
3231:(
3221:)
3213:G
3210:E
3204:)
3201:s
3198:,
3193:M
3188:(
3183:(
3158:)
3151:)
3140:)
3135:i
3131:s
3127:,
3122:M
3117:(
3112:(
3107:i
3101:)
3096:1
3092:s
3088:=
3085:s
3082:(
3068:2
3064:s
3055:1
3051:s
3039:(
3029:)
3021:G
3018:A
3012:)
3009:s
3006:,
3001:M
2996:(
2991:(
2966:)
2959:)
2948:)
2943:1
2939:s
2935:,
2930:M
2925:(
2920:(
2910:1
2906:s
2899:s
2888:(
2878:)
2870:X
2867:E
2861:)
2858:s
2855:,
2850:M
2845:(
2840:(
2815:)
2808:)
2797:)
2792:1
2788:s
2784:,
2779:M
2774:(
2769:(
2759:1
2755:s
2748:s
2737:(
2727:)
2719:X
2716:A
2710:)
2707:s
2704:,
2699:M
2694:(
2689:(
2664:)
2657:)
2650:)
2643:2
2632:)
2629:s
2626:,
2621:M
2616:(
2611:(
2598:)
2591:1
2580:)
2577:s
2574:,
2569:M
2564:(
2559:(
2549:(
2539:)
2532:)
2525:2
2514:)
2511:s
2508:,
2503:M
2498:(
2493:(
2483:)
2476:1
2465:)
2462:s
2459:,
2454:M
2449:(
2444:(
2437:(
2430:(
2420:)
2413:2
2400:1
2389:)
2386:s
2383:,
2378:M
2373:(
2368:(
2343:)
2336:)
2329:2
2318:)
2315:s
2312:,
2307:M
2302:(
2297:(
2287:)
2280:1
2269:)
2266:s
2263:,
2258:M
2253:(
2248:(
2241:(
2231:)
2224:2
2211:1
2200:)
2197:s
2194:,
2189:M
2184:(
2179:(
2154:)
2147:)
2140:2
2129:)
2126:s
2123:,
2118:M
2113:(
2108:(
2098:)
2091:1
2080:)
2077:s
2074:,
2069:M
2064:(
2059:(
2052:(
2042:)
2035:2
2022:1
2011:)
2008:s
2005:,
2000:M
1995:(
1990:(
1965:)
1958:)
1951:2
1940:)
1937:s
1934:,
1929:M
1924:(
1919:(
1909:)
1902:1
1891:)
1888:s
1885:,
1880:M
1875:(
1870:(
1863:(
1853:)
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